A Fast and Elitist Multiobjective Genetic Algorithm: NSGA

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A Fast and Elitist Multiobjective Genetic
Algorithm NSGA-?

• K DEBL, A PRATAP, S AGRAW

• Ying-Shiuan You(???)

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Multi-objective Optimization

• The process of simultaneously optimizing two or
  more conflicting objectives subject to certain
  constraints.
• Performance vs. Cost
• Solution Methods
• Constructing a single aggregate objective
  function(AOF).
• e.g.
  (weighted linear sum)
• Hard to specify weight.
• Multiobjective Optimization Evolutionary
  Algorithms (MOEA).
• etc.

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Pareto Optimality

• Pareto Improvement
• A change in the status such that makes at least
• one dimension better off, while making no
  other
• dimension worse off.
• Pareto Optimal
• No further Pareto improvements can be made.
• Dominance
• Solution dominates another if no worse in all
  objectives and strictly better than on one.

Vilfredo Pareto (18481923)
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Pareto Front

• Produce non-dominated set of solutions with
  regard to all objectives.
• All solutions on the pareto front is optimal.

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Criticisms of the NSGA

• High computational complexity of nondominated
  sorting.
• O(MN3), M objectives, N population size.
• Lack of elitism.
• Elitism would speed up the performance of GA.
• Need for specifying the sharing parameter
  .
• Hard to specify.

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Naive Nondominated Sorting Approach

• Worst case N fronts. -gt

Find first nondominated front.
N solution
Compare each other solution.
Find other front.
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Fast Nondominated Sorting Approach

• Find front time , space .
• Total time complexity

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Diversity Preservation

• Crowding in objective space.
• Crowding distance perimeter of the cuboid
  formed by nearest neighbors.
• Smaller Crowding distance -gt denser in the
  neighborhood.

Crowding-distance-assignment
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Diversity Preservation

• Crowded-Comparison Operator
• 1) Nondomination rank(irank)
• 2) Crowding distance(idistance),
• Partial order

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Elitist Replacement

• Combine parent and offspring population.
• Select better ranking individuals and use
  crowding distance to break the tie.

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Main Loop

• Complexity
• Nondominated sorting O(M(2N)2)
• Crowding-distance assignment O(M(2N)log(2N))
• Sorting on O(2Nlog(2N))

• Initialize population.
• Each solution is assigned a fitness equal to its
  nondomination level.
• Selection, recombination, mutation.
• Elitist Replacement.

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Experiment Problem
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Experiment Result
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Experiment Result (contd)
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Constraint Handling

• Proposed Constraint-Handling Approach
• Definition 1 A solution i is said to
  constrained-dominate a solution j, if any of the
  following conditions is true.
• i is feasible and j is not.
• i and j are both infeasible, but i has a smaller
  overall constraint violation.
• i and j are feasible and i dominates j.

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Constraint Handling (contd)

• Ray-Tai-Seows Constraint-Handling Approach
• First rank M objective function. Robj
• Second rank violation value of J constraint.
  Rcon
• Third rank combination of above two. Rcom
• All feasible solution having best rank at Rcom
  chosen for new population.
• If need more individual, generate systematically.
• importance on Robj in selection and importance
  on Rcon in XO.
• Objective value and constraint value used
  together. -gt No need of penalty parameter.

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Experiment Result
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Contribution

• Proposed a computationally fast and elitist MOEA.
• Proposed a simple extension to the definition of
  dominance for constrained multi-objection
  optimization.